Hopf Galois module structure of dihedral degree 2p extensions of ℚ_p

2020/11/10 by Daniel Gil-Muñoz, Gil-Muñoz, Daniel, Anna Rio +1
Mathematics · #11R32 #12F10 #16T05 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2011.05024

openalex publication_date 2020/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be an odd prime. For field extensions L/ℚ_p with Galois group isomorphic to the dihedral group D_2p of order 2p, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and the module structure of the ring of integers O_L. We solve the case in which L/ℚ_p is not totally ramified and present a practical method which provides a complete answer for the cases p=3 and p=5. We see that within this family of dihedral extensions, the ring of integers is always free over the associated orders in the different Hopf Galois structures.

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